An application of the inequality for modified Poisson kernel
- Gaixian Xue^{1} and
- Junfei Wang^{2}Email author
https://doi.org/10.1186/s13660-016-0959-6
© Xue and Wang 2016
Received: 14 September 2015
Accepted: 4 January 2016
Published: 22 January 2016
Abstract
As an application of an inequality for modified Poisson kernel obtained by Qiao and Deng (Bull. Malays. Math. Sci. Soc. (2) 36(2):511-523, 2013), we give the generalized solution of the Dirichlet problem with arbitrary growth data.
Keywords
1 Introduction and results
Let \(\mathbf{R}^{n}\) (\(n\geq2\)) be the n-dimensional Euclidean space. The boundary and the closure of a set E in \(\mathbf{R}^{n}\) are denoted by ∂E and E̅, respectively. The Euclidean distance of two points P and Q in \(\mathbf{R}^{n}\) is denoted by \(\vert P-Q\vert \). Especially, \(\vert P\vert \) denotes the distance of two points P and O in \(\mathbf{R}^{n}\), where O is the origin in \(\mathbf{R}^{n}\).
We introduce a system of spherical coordinates \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \(\mathbf{R}^{n}\) which are related to Cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).
Let \(B(P,r)\) denote the open ball with center at P and radius r (>0) in \(\mathbf{R}^{n}\). The unit sphere and the upper half unit sphere in \(\mathbf{R}^{n} \) are denoted by \(\mathbf{S}^{n-1}\) and \(\mathbf{S}_{+}^{n-1}\), respectively. The surface area \(2\pi^{n/2}\{\Gamma(n/2)\}^{-1}\) of \(\mathbf{S}^{n-1}\) is denoted \(w_{n}\). Let \(\Omega\subset\mathbf{S}^{n-1}\), a point \((1,\Theta)\) and the set \(\{\Theta; (1,\Theta)\in\Omega\}\) are denoted Θ and Ω, respectively. For two sets \(\Lambda\subset\mathbf{R}_{+}\) and \(\Omega\subset\mathbf{S}^{n-1}\), we denote \(\Lambda\times\Omega=\{ (r,\Theta)\in\mathbf{R}^{n}; r\in\Lambda,(1,\Theta)\in\Omega\}\), where \(\mathbf{R}_{+}\) is the set of all positive real numbers.
For the set \(\Omega\subset\mathbf{S}^{n-1}\), we denote the set \(\mathbf{ R}_{+}\times\Omega\) in \(\mathbf{R}^{n}\) by \(C_{n}(\Omega)\), which is called a cone. For the set \(I\subset \mathbf{R}\), the sets \(I\times\Omega\) and \(I\times\partial{\Omega}\) are denoted \(C_{n}(\Omega;I)\) and \(S_{n}(\Omega;I)\), respectively, where R is the set of all real numbers. Especially, the set \(S_{n}(\Omega; \mathbf{R}_{+})\) is denoted \(S_{n}(\Omega)\).
Recently, Qiao and Deng (cf. [1]) gave the solution of the Dirichlet problem in a cone. Applications of modified Poisson kernel with respect to the Schrödinger operator, we refer the reader to the papers by Huang and Ychussie (see [6]) and Li and Ychussie (see [7]).
Theorem A
Furthermore, Qiao and Deng (cf. [4]) supplemented the above result and proved the following.
Theorem B
As an application of the inequality (1) and the generalized Poisson kernel \(PI_{\Omega}^{\phi}(P,Q)\), we have the following.
Theorem
2 Lemmas
Lemma 1
Proof
Lemma 2
(See [4])
Suppose that the following two conditions are satisfied:
3 Proof of Theorem
To see that \(H_{\Omega}^{\phi_{g}}(P)\) is a harmonic function in \(C_{n}(\Omega)\), we remark that \(H_{\Omega}^{\phi_{g}}(P)\) satisfies the locally mean-valued property by Fubini’s theorem.
For any \(\epsilon>0\) and a positive number δ, by (9) we can choose a number R (\(>\max\{1,2(t'+\delta)\}\)) such that (6) holds, where \(P\in C_{n}(\Omega)\cap B(Q',\delta)\).
Thus we complete the proof of Theorem.
Declarations
Acknowledgements
The authors are very thankful to the anonymous referees for their valuable comments and constructive suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
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